I have a finite measure space $(X, \mathcal{S}, \mu)$, and a transformation $f:X\rightarrow X$ that "preserves measure on average". That is, for $A \in \mathcal{S}$ $$ \lim_{n\rightarrow \infty} \frac{1}{n}\sum_{k=1}^n \mu(f^k(A)) = \mu(A) $$ I would like to be able to apply some ergodic or recurrence theorem in this situation, but Poincare Recurrence for example requires that the map $f$ be measure-preserving, which this "measure-preserving on average" mapping does not satisfy.

I am wondering if anyone can say if there are any results concerning such "measure-preserving on average" maps and any related recurrence or ergodic theorems, or if there is some way I can use say an altered Poincare Recurrence to makes claims about this mapping.

isinvariant: you can just substitute $f^{-1}(A)$ for $A$ and you see you have the same limit, so your condition implies $\mu(A)=\mu(f^{-1}A)$. $\endgroup$ – Anthony Quas Dec 7 '20 at 2:36